The eigenvalue equation on the Eguchi-Hanson space

Eguchi-Hanson singularities in U(2)-invariant Ricci flow

We show that a Ricci flow in four dimensions can develop singularities modeled on the Eguchi-Hanson space. In particular, we prove that starting from a class of asymptotically cylindrical $U(2)$-invariant initial metrics on $TS^2$, a Type II singularity modeled on the Eguchi-Hanson space develops in finite time. Furthermore, we show that for these Ricci flows the only possible blow-up limits are (i) the Eguchi-Hanson space, (ii) the flat $\mathbb{R}^4 /\mathbb{Z}_2$ orbifold, (iii) the 4d Bryant soliton quotiented by $\mathbb{Z}_2$, and (iv) the shrinking cylinder $\mathbb{R} \times \mathbb{R}…

Gluing Eguchi-Hanson metrics and a question of Page

In 1978, Gibbons-Pope and Page proposed a physical picture for the Ricci flat Kähler metrics on the K3 surface based on a gluing construction. In this construction, one starts from a flat torus with $16$ orbifold points, and resolves the orbifold singularities by gluing in $16$ small Eguchi-Hanson manifolds which all have the same orientation. This construction was carried out rigorously by Topiwala, LeBrun-Singer, and Donaldson. In 1981, Page asked whether the above construction can be modified by reversing the orientations of some of the Eguchi-Hanson manifolds. This is a subtle question: if…

源自Kähler–Einstein边缘度量的Eguchi–Hanson度量

Eguchi--Hanson metrics arising from Kahler--Einstein edge metrics

本文在Calabi–Hirzebruch流形上构造沿两个除子奇异的Kähler–Einstein边缘度量，研究锥角取极值时的Gromov–Hausdorff极限，证明Eguchi–Hanson度量自然出现，并完整描述所有极限行为，解决了Cheltsov–Rubinstein猜想。

Eguchi-Hanson 调和旋量再探

Eguchi-Hanson harmonic spinors revisited

本文重新探讨了 Eguchi-Hanson 空间上的 Dirac 算子零模问题，利用 spin-$c$ 旋量形式体系简化了计算，无需计算旋量联络即可重现扭曲 Dirac 算子的可归一化零模，并将结果推广至更广泛的 Ricci-flat Kähler 流形。

对Calabi-Eguchi-Hanson空间的详细考察

A detailed look at the Calabi-Eguchi-Hanson spaces

本文详细研究了复射影空间标准线丛上的Calabi-Eguchi-Hanson Ricci平坦度量，将其刻画为orbifold奇点的解消，并通过显式计算曲率证明了所有紧致极小子流形均位于零截面中，推广了Tsai和Wang的结果。

Infinitely many new families of complete cohomogeneity one G_2-manifolds: G_2 analogues of the Taub-NUT and Eguchi-Hanson spaces

We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G_2-ma…

Irregular Eguchi-Hanson type metrics and their soliton analogues

We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled S…

A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces

We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $ι$ preserving the $G_2$-structure. Then $M/{\langle ι\rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $λ$ on $L$. Much of…

Geometric and analytic properties of families of hypersurfaces in Eguchi-Hanson space

We study the geometry of families of hypersurfaces in Eguchi-Hanson space that arise as complex line bundles over curves in $S^2$ and are three-dimensional, non-compact Riemannian manifolds, which are foliated in Hopf tori for closed curves. They are negatively curved, asymptotically flat spaces, and we compute the complete three-dimensional curvature tensor as well as the second fundamental form, giving also some results concerning their geodesic flow. We show the non-existence of $Ł^p$-harmonic functions on these hypersurfaces for every $p \geq 1$ and arbitrary curves, and determine the infi…